Stochastic Integration on Stochastic Sets of Interval Type and Applications to Mathematical Finance

TL;DR

This paper develops new types of stochastic integrals on stochastic sets of interval type, extending stochastic analysis and applying it to financial models with uncertain time horizons, allowing for better filtering of asset dynamics.

Contribution

It introduces three novel $$-stochastic integrals, linking them to existing integrals and deriving Itô's formula within this framework, with applications to finance models with uncertain horizons.

Findings

Defined $$-stochastic integrals for various processes

Established relationships with classical stochastic integrals

Applied to financial markets with sudden-stop horizons

Abstract

In the existing works, stochastic sets $\mathbb{B}$ of interval type, along with $\mathbb{B}$-stochastic processes, were introduced within the framework of stochastic analysis. In this paper, we undertake the construction of $\mathbb{B}$-stochastic integration by exploring three novel types of $\mathbb{B}$-stochastic integrals: Stieltjes integrals of $\mathbb{B}$-predictable processes with respect to $\mathbb{B}$-adapted processes with finite variation, stochastic integrals of $\mathbb{B}$-predictable processes with respect to $\mathbb{B}$-inner local martingales, and stochastic integrals of $\mathbb{B}$-predictable processes with respect to $\mathbb{B}$-inner semimartingales. These $\mathbb{B}$-stochastic integrals are exclusively defined on subsets $\mathbb{B}$, with values outside the scope of $\mathbb{B}$ being deemed irrelevant. Additionally, we present several notable consequences, including the relationship between $\mathbb{B}$-stochastic integrals and existing stochastic integrals, as well as It\^{o}'s formula for $\mathbb{B}$-inner semimartingales. In the context of models pertaining to uncertain time-horizons in mathematical finance, we establish essentials of mathematical finance for general markets characterized by sudden-stop horizons. This is achieved by defining self-financing strategies, admissible strategies, and no-arbitrary conditions. In such financial markets, the exclusivity characteristic inherent in $\mathbb{B}$-stochastic integrals offers investors a viable alternative approach. This approach enables them to effectively filter out unnecessary information pertaining to asset price dynamics and portfolio strategies that extend beyond the predefined time-horizons.